Making Fractions 1

Task 222 ... Years 4 - 10

Summary

Students first arrange a set of blocks in order and record the order on the board. The purpose of biggest to smallest ordering is to make more obvious a relationship that connects the pieces when the puzzle is solved. The puzzle is based on value relations. That is, if we give this piece a certain value, what is the value of each of the other related pieces. The assigned value is always 1, in other words, the assigned piece becomes the whole for that row of the puzzle. Other pieces will be related to the whole as fractions or mixed numbers and the first challenge is to find these relationships in each row. The challenge section of the card introduces decimal equivalents by giving the chosen piece the value of $1.

Materials

One set of 27 blocks in 5 sizes - each size has a fraction relationship to the others

Recording board, marking pen and cloth

Content

arithmetic, multiplication / division

concept of proof

decimals, calculations

fractions, calculations

fractions, equivalence

fractions, reciprocal

fractions, whole & parts

fractions, value relations

measurement, area

mental arithmetic

money

spatial perception, 2D or 3D

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The principle of this task is to take a whole, divide into parts, then choose any of those parts in turn to have a value of 1. In effect, the task is based on the concept that any piece, not necessarily the largest one, can be the whole. Making Fractions 1 is based on quarters, fifths, eighths and ninths of the largest square.

To record the biggest to smallest order along the top line of the chart, students can use drawings, or a code such as LS for largest rectangle. The pieces are not intended to fit in these cells. However, some students do stand them on their edge outside the cells. An additional spatial challenge is provided by trying to fit the pieces back into the frame at pack up time. This should be encouraged because it is also a check that all the pieces are there.

From largest to smallest the blocks are:

The direct relationships between the blocks are limited, however a full set of each shape does have the same area as the largest square and therefore as each other. We are now forced into proportional reasoning. For example:

Four Bs has the same area as five Cs.

So one B must have more area than one C.

If B is worth 1, then each C must be four fifths of B.

If C is worth 1, then each B must be five fourths (1 1/4) of C.

Can this reasoning be checked another way?
Hint: Explore the reasoning about stacking blocks in the cameo for Making Fractions 3.

When correctly filled in the table is:

A

B

C

D

E

Row 1

1

1/4

1/5

1/8

1/9

Row 2

4

1

4/5

1/2

4/9

Row 3

5

11/4

1

5/8

5/9

Row 4

8

2

13/5

1

8/9

Row 5

9

21/4

14/5

21/8

1

Encourage students to look for patterns in the table. With a little prodding they may see, especially if their attention is drawn to the first row and column, that cells reflected in the leading diagonal are reciprocals of each other - that is, the product of the two cell values is 1.

Examples
Row 2E (4/9) and Row 5B (9/4) could be thought of as:

4/9 of 21/4

21/4 lots of 4/9

4/9 x 9/4

9/4 x 4/9

but in whichever way the answer is 1.

Challenge
The card also asks students to try the same problem again by choosing the value of $1 for each piece in turn. This problem relates the fraction question to decimals (or money). It also involves making a decision about what to do with parts of a cent in prices like 1/9 of a dollar. Should these be expressed as decimals (rounded where appropriate) or would the manufacturer of the pieces leave the figure in the fraction form on the grounds that pieces would be sold in large quantities?Note: There are times, such as in currency exchange, that fractions of a cent are used ... and these small bits matter a lot if you are exchanging large amounts of currency!

If fractions are used, the table becomes:

A

B

C

D

E

Row 1

$1

25¢

20¢

121/2¢

111/9¢

Row 2

$4

$1

80¢

50¢

444/9¢

Row 3

$5

$1·25

$1

621/2¢

555/9¢

Row 4

$8

$2

$1·60

$1

888/9¢

Row 5

$9

$2·25

$1.80

$1·121/2

$1

How do the reciprocal relationships across the leading diagonal work out now?

What would the table look like if it were all in 'currency exchange' form, rounded if appropriate? What would happen to the reciprocal relationships?

More Fraction Calculations

Fractions are all about knowing what the whole is, dividing the whole into equal parts and then, as a consequence of the partitioning, being in a position to choose and use the appropriate fraction language. For example: Five of these equal parts make the whole so each part is ...worth one fifth (...called one fifth).

Now, other fraction statements (stories? / equations?) are obvious from the pieces such as:

one fifth + one fifth + one fifth + one fifth + one fifth = 1

1 - three fifths = two fifths

5 x one fifth = 1

four fifths ÷ (how many) one fifth(s) = 4

and it is better to encourage writing what is said (one fifth), rather than symbols until the students can tell you that they know enough to use the symbols and explain them.

If that same whole can be divided into equal parts in another way, then a different fraction word comes into play and equivalent fractions are possible. However, in this Fraction Set the only directly connected shapes apart from A which connects to them all are B and D.

If A is the whole, B is one fourth and C is one fifth.
But how do you evaluate one fourth + one fifth with this set?

The answer is that you can't. You have to invent another way of dividing the whole to make a new shape which is common to both the fourths and the fifths. Exploration of this type of question could be the basis of a teacher created Investigation Guide. When you write one and have trialled and modified it, please submit it to be shared with colleagues.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

To convert this task to a whole class investigation each pair (or perhaps group of 4) will need a set of blocks. Multiple copies of the task, or additional panels of blocks can be purchased from Mathematics Centre, or something similar can be made from wood or card. However, these approaches might place an excess demand on finances or time. Instead, consider using a work station approach where this task and several others with similar mathematics as suggested above in the Content Finder, are one station. The other work stations might be text based work on value relations and software involving value relations, such as that for the Maths300 companion lesson of Task 75, What's It Worth?. With a system like this, where the text and software stations tend to 'look after themselves' the teacher can often find time to spend with the task group to listen to, question and assess their mathematical discussion.

At this stage, Making Fractions 1 does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.